Problem: Captain Ben has a ship, the H.M.S Crimson Lynx. The ship is five furlongs from the dread pirate Luis and his merciless band of thieves. If his ship hasn't already been hit, Captain Ben has probability $\dfrac{2}{3}$ of hitting the pirate ship. If his ship has been hit, Captain Ben will always miss. If his ship hasn't already been hit, dread pirate Luis has probability $\dfrac{1}{3}$ of hitting the Captain's ship. If his ship has been hit, dread pirate Luis will always miss. If the Captain and the pirate each shoot once, and the pirate shoots first, what is the probability that the pirate hits the Captain's ship, but the Captain misses?
Solution: The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened. In this case, event A is the pirate hitting the Captain's ship and event B is the Captain missing the pirate ship. The pirate fires first, so his ship can't be sunk before he fires his cannons. So, the probability of the pirate hitting the Captain's ship is $\dfrac{1}{3}$. If the pirate hit the Captain's ship, the Captain has no chance of firing back. So, the probability of the Captain missing the pirate ship given the pirate hitting the Captain's ship is $1$. The probability that the pirate hits the Captain's ship, but the Captain misses is then the probability of the pirate hitting the Captain's ship times the probability of the Captain missing the pirate ship given the pirate hitting the Captain's ship. This is $\dfrac{1}{3} \cdot 1 = \dfrac{1}{3}$